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Questions tagged [mathematical-foundations]

Questions about the relation of (subfields of) computer science to the relevant mathematical foundations and their application.

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2 answers
69 views

Mathematical Foundation for Dimensioned Quantities?

Is there support in the foundation of mathematics for dimensioned quantities? Those immersed in real-world software problems, continually confront the absence of dimensioned quantities in programming ...
2 votes
1 answer
97 views

Is the root function computationally equivalent to function application?

If a function type is representable by exponentiation, does it follow that function application is represented by the right inverse, roots? It would seem that roots consume a function's input to ...
1 vote
1 answer
194 views

Translating the in-order index of a node in a complete, balanced binary tree into the level-order index

Consider the topmost part of a complete, balanced binary tree of all 64-bit numbers, exemplified here. As highlighted by the lack of a 7*2^64/8 term it is not ...
10 votes
3 answers
1k views

Resources for studying the mathematical foundations of machine learning, for someone from a math/physics background

I am a soon-to-be physics graduate student with a background in theoretical and experimental cosmology. In my work, I've often found myself applying machine learning models and techniques for the ...
0 votes
2 answers
117 views

How to represent a point cloud in the pseudocode of an algorithm?

I am writing a scientific paper in which I describe some algorithms (using pseudocode) that have point clouds as inputs. In these algorithms, I need a mathematical structure to represent a point cloud....
1 vote
0 answers
69 views

How exactly does minimizing the L2 norm of w in SVR affect the model's ability to fit the data?

I have a solid grasp on why we minimize the L2 norm of the weight vector ( w ) in Support Vector Machines (SVM) for classification problems — it maximizes the margin between classes and helps the ...
5 votes
7 answers
3k views

What are the conditions necessary for a programming language to have no undefined behavior?

For context, yesterday I posted Does the first incompleteness theorem imply that any Turing complete programming language must have undefined behavior?. Part of what prompted me to ask that question ...
0 votes
1 answer
153 views

Is "Bitwise Complement Operator" (~ tilde) distributive?

To be more precise, Is ~(a+b) = ~a + ~b? Here, "~" bitwise NOT operator. I ran into this question while thinking about ...
1 vote
2 answers
119 views

Division by zero in APL: assign return value 0 or 1

Recently I began learning a couple array based programming languages: Dyalog APL and BQN. And I cam across this peculiar manner in which division by zero is handled in Dyalog APL. Using the in-built ...
-1 votes
1 answer
55 views

How to show invariance under additive and multiplicative illumination changes

The task is to "Show that the transformation f -> ((ln fx)x, (ln fy)y) is invariant under additive and multiplicative illumination changes." From my lecture notes, for ((ln f)x, (ln f)y), ...
219 votes
32 answers
53k views

Why is writing down mathematical proofs more fault-proof than writing computer code?

I have noticed that I find it far easier to write down mathematical proofs without making any mistakes, than to write down a computer program without bugs. It seems that this is something more ...
10 votes
4 answers
6k views

Could Gödel’s incompleteness theorem be circumvented with a quine?

As you all probably know, Gödel’s incompleteness theorem states, that it will never be possible for mathematics to prove its own correctness. Mainly because that proof would be part of mathematics too,...
2 votes
0 answers
41 views

Reading list for mathematics and formal logic with missing truth values?

I would like a reading list of math and formal logic books which give a principled discussion of missing values. Textbooks on mathematics and formal logic - propositional and first-order and higher-...
1 vote
0 answers
79 views

Mathematical resource material accompanying TAPL

I'm currently reading Types and Programming Languages by Benjamin C. Pierce and just arrived at chapter 21 Metatheory of Recursive Types. Prior to this chapter I found the book challenging but ...
0 votes
1 answer
47 views

How to evaluate all the binary sequences, generated from $2^{100}$ for finding all the sequeces which contain minimum $10$ zeros?

Suppose I have a set of $2^{n}$ number of binary sequences. And I have to select only those sequences which contain a minimum ${P}$ number of $0$ in it. For example, please consider the below one Eg. ...
1 vote
0 answers
53 views

Finding square root of a gram matrix over the integers [closed]

Suppose that matrix A is a symmetric positive definite matrix over the integers, i.e., $A \in Z^{n\times n}$, if B is a matrix over the real numbers, it is not difficult to find B such that $A = B \...
2 votes
1 answer
88 views

Why is this mapping from binary to natural numbers surjective?

In my Lecture we came across something of importance for the Church-Turing-Thesis and i noticed one particular function, which confused me. Function: Let $bin(x)$ be the injective binary extension ...
1 vote
0 answers
166 views

What algorithm is best for resource gathering in an RTS game in quickest time?

Though the concept can be for many RTS games but lets use the specific system of 'age of empires'. For simplicity one resource will be considered. Food. So in the game there are villagers who gather ...
-1 votes
1 answer
56 views

What does x sign mean in functions

I came across this cross sign when reading a book. Does anyone know what does this mean in the context of a function?
4 votes
3 answers
406 views

How Types avoids Russel's Paradox?

I gone through the Russel's paradox. From Wikipedia : According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is ...
5 votes
1 answer
111 views

Are the sets in $\Sigma_{p}$ (or $\Pi_{p}$) totally ordered?

Are the sets in each stage of the arithmetic hierarchy well-ordered, with respect to : $T-$reductions, or $m-$reductions? It is something which I have been unclear with for a while (from a ...
6 votes
2 answers
1k views

A general picture of formal verification in software

I'm trying to piece together a general picture of the state of formal software verification, and I'm having a good bit of trouble. For context, I come from mostly a math background. I'm quite familiar ...
0 votes
2 answers
333 views

Tight bound on the number of divisors of n

The inner loop of this pseudocode runs only for the divisors of n for i=1 to n: if(n%i==0): for j=1 to n: Do something in constant time If I ...
0 votes
1 answer
62 views

Is a zero matrix nilpotent matrix?

Let a matrix $A=0$. Can I say $A$ is nilpotent? I am asking if the zero matrix is nilpotent or not. Nilpotent means that $A^k = 0$ for some $k$ that is a non-negative integer. When $k = 1$, $A^k = A = ...
1 vote
1 answer
74 views

Efficient cardinality of set overlap relation

Assume that we have a set S of sets s. Every pair (s,s') in ...
2 votes
2 answers
1k views

Is discrete math enough for computer science ? Or there other Math topics that I should also learn With it?

I want to learn computer science, SO is discrete math enough for computer science ? Or there other Math topics that I should also learn With it ? I don’t have specific topic that I care more about ...
2 votes
0 answers
32 views

How to specify mutated types mathematically?

Say I have an object which I pass to a method, and the method returns that same object, just mutated. So it goes like this: ...
5 votes
3 answers
762 views

Usefulness of Differential Geometry

I recently came across these books: Differential Geometry and Lie Groups: A Computational Perspective Differential Geometry and Lie Groups: A Second Course Their subject matter really intrigues me, ...
0 votes
0 answers
64 views

Function notation/mapping for this type of functions?

In short, what is the proper function notation/mapping for this type of functions? From Wikipedia regarding maps: In the communities surrounding programming languages that treat functions as first-...
0 votes
0 answers
54 views

Prove that for any finite k = 1, 2, 3 … PCPk is decidable

Let PCPk be a version of PCP where we can use at most k copies of each domino to create correspondences. Prove that for any finite k = 1, 2, 3 … PCPk is decidable. I know this kind of PCP is called ...
1 vote
1 answer
22 views

Proof the number of expected node in a binary using mathematical induction

The following algorithm constructs a binary tree. ...
0 votes
0 answers
30 views

Papers on depth separations in neural networks

I am new to the domain of Machine learning. I have been asked to present a paper related to the mathematics behind the depth separations in Neural Networks (by Itay Safran, Ronen Eldan and Ohad Shamir)...
0 votes
0 answers
65 views

A set that is not polynomial time enumerable

For a set $I\subseteq \Bbb N$, defined $s_{I}(n)=\min\{i\in I\mid i>n\}$. The set $I$ is called polynomial-time-enumerable if $s_I(n)$ is computable in $\mathsf{poly}(n)$ time. Most of the sets I ...
11 votes
1 answer
2k views

Monad in Haskell programming vs. Monad in category theory

I have a question about concept of monad used in Haskell programming and category theory in math. Recall in Haskell a monad consists of following components: A type constructor that defines for each ...
1 vote
1 answer
82 views

"The Annotated Turing" on listing all binary numbers between 0 and 1

In his book "The Annotated Turing" in the first sentence on page 32 Charles Petzold wrote: These are binary numbers between 0 and 1, and (judging from the way we created these numbers) all ...
8 votes
2 answers
720 views

What are the implications of Homotopy Type Theory?

I've recently come across the topic of homotopy type theory and I'm interested to learn more. I have a very limited background in type theory. Can anyone tell me, in functional programming terms or ...
-1 votes
3 answers
54 views

Which topics of mathematics should I need to learn to be a good app developer?

I'm 29 years old. I couldn't continue my studies after grade 10 due to some financial issues and I didn't have time to practice mathematics. It's been more than 11 years since I left studies. Now I ...
7 votes
1 answer
212 views

What is a list of "reasonable but undecidable" theorems?

There are some theorems that go along the lines of "all reasonable properties of <math subject> are computationally undecidable." Here are two examples: Rice's Theorem: "all ...
0 votes
1 answer
61 views

Mathematical expression for the quantity that we are maximising in the stock buying and selling problem

Problem Statement: Say you have an array prices for which the $i^{th}$ element is the price of a given stock on day $i$. Design an algorithm to find the maximum profit. You may complete as many ...
1 vote
2 answers
106 views

A silly doubt on Computer Science Objects: Arrays, Lists, Strings, and Vectors, are they the same concept?

First of all I'm a Physics Student and I've been arguing with my brother concerning the usage and the fundamental meaning of a vector on Computer Science (CS). I) Vectors A vector have a axiomatic ...
1 vote
1 answer
1k views

What is the difference between a fraction and a float?

I understand any fraction to be a quotient of integers which isn't 0, but after coming across the term "float" in various programming languages (such as JavaScript) I misunderstand why it is ...
1 vote
1 answer
94 views

How was the σ function chosen to extend the perceptron?

I am just reading about perceptrons in more depth, and now onto Sigmoid Neurons. Some quotes: A small change in the weights or bias of any single perceptron in the network can sometimes cause the ...
2 votes
4 answers
231 views

How does maths consistency impacts on computer science?

Lot of mathematicians have made several efforts to answer the question : are mathematics consistent ? Although we haven't yet had a proof of consistency, and surely we will never (Gödel second theorem)...
0 votes
1 answer
46 views

How do you convert bits into a different alphabet?

I have forgotten how to do this. How do I figure out what the requirements are for a 128-bit string using a certain alphabet? That is to say, I want to generate a UUID (128-bit) value, using only the ...
0 votes
1 answer
260 views

Sets in Mathematics are immutable but in Computer Science sets are mutable and called "Dynamic Sets" - truth of the statement

While reading the classic text Introduction to Algorithms by Cormen et. al. I came across the following claim: Sets are as fundamental to computer science as they are to mathematics. Whereas ...
2 votes
1 answer
1k views

How is expected value in entropy derived?

I was self learning about entropy and came across this equation. $$ H = - \sum p(x) \log p(x) $$ The equation for entropy in expected value, $$ H(x) = \operatorname*{\mathbb{E}}_{X \sim P}[I(x)] = -\...
0 votes
1 answer
69 views

Big O Tires Question? [duplicate]

My question is regarding the last paragraph of this excerpt from "Cracking the Coding Interview." (For some reason, my table is not formatting here.) What's the runtime of this code? ...
1 vote
3 answers
388 views

Prove little o with just the definition

I have been searching for a while now but couldn't find anything about this exact pair of functions with the little $\mathcal{o}$ notation. Given the functions $f(n) = 2^{n}$ and $g(n) = n!$ I am ...
0 votes
1 answer
87 views

Where does the base come from in a subsequence count?

If sequence X has m symbols, then X has 2m possible subsequences. Where does the base (<...
3 votes
1 answer
776 views

Why is addition in GF(2^8) the same as XOR?

I get the impression it has to do with either some quirk involved with limiting to 2^8 or that I'm misunderstanding what addition can be within the context of a finite field, but I'm not quite sure ...